The Kolmogorov-Centsov theorem and Brownian motion in vector lattices

The well known Kolmogorov-Centsov theorem is proved in a Dedekind complete vector lattice (Riesz space) with weak order unit on which a strictly positive conditional expectation is defined. It gives conditions that guarantee the Holder-continuity of a stochastic process in the space. We discuss the notion of independence of projections and elements in the vector lattice and use this together with the Kolmogorov-Centsov theorem to give an abstract definition of Brownian motion in a vector lattice. This definition captures the fact that the increments in a Brownian motion are normally distributed and that the paths are continuous. (C) 2013 Elsevier Inc. All rights reserved.